Three

My first reaction to this extensive autobiographical account was one of admiration. Uncle Petros had given me the facts of his life with remarkable honesty. It wasn't until a few days later, when the oppressive influence of his melancholy narrative diminished, that I realized everything he'd told me had been beside the point.

Remember that our meeting had been initially arranged so that he could try to justify himself. His life's story was only relevant to the extent that it explained his atrocious behaviour, assigning me in all my adolescent mathematical innocence the task of proving Goldbach's Conjecture. Yet, during his long narrative he had not even touched on his cruel prank. He'd ranted on and on about his own failure (or maybe I should do him the favour of calling it 'bad luck'), but about his decision to turn me away from studying mathematics and the method he had chosen to implement it, not a single word. Did he expect me automatically to draw the conclusion that his behaviour to me was determined by his own bitter life-experiences? It didn't follow: although his life story was indeed a valid cautionary tale, it taught a future mathematician what pitfalls to avoid so as to make the most of his career – not how to terminate it.

I let a few days go by before I went back to Ekali and asked him point-blank: could he now explain why he had attempted to dissuade me from following my inclination.

Uncle Petros shrugged. 'Do you want the truth?'

'Of course, Uncle.' I said. 'What eise?'

'All right then. I believed from the first moment – and still do, I'm sorry to say – that you have no special gift for great mathematics.'

I became, once again, furious. 'Oh? And how on earth could you have known that? Did you ask me a single mathematical question? Did you ever set me a problem to solve, other than the unprovable, as you termed it, Conjecture of Christian Goldbach? I certainly hope you don't have the nerve to tell me that you deduced my lack of mathematical ability from that!'

He smiled, sadly. 'You know the popular saying that the three conditions impossible to conceal are a cough, wealth and being in love? Well, to me there is a fourth: mathematical gift.'

I laughed contemptuously. 'Oh, and you can no doubt identify it at a glance, eh? Is it a look in the eye or a certain je ne sais quoi that betrays to your ultra-fine sensibility the presence of mathematical genius? Can you perhaps also determine one's IQ with a hand-shake?'

'Actually there is an element of that "look in the eye",’ he replied, ignoring my sarcasm. 'But in your case physiognomy was only a small part of it. The necessary – but not sufficient, mind you – precondition for supreme achievement is single-minded devotion. If you had the gift that you yourself would like to have had, dear boy, you wouldn't have come asking for my blessing to study mathematics; you would have gone ahead and done it. That was the first tell-tale sign!'

The more he explained himself, the angrier I got. 'If you were so certain I wasn't gifted, Uncle, why did you put me through the horrific experience of that summer? Why did I have to be subjected to the totally unnecessary hurmliation of thinking myself a near-idiot?'

'But, don't you see?' he answered merrily. 'Goldbach's Conjecture was my security! If by some remote chance I'd been wrong about you and, in the most unlikely instance, you were indeed earmarked for greatness, then the experience wouldn't have crushed you. In fact it would not have been at all "horrific", as you significantly termed it, but exciting and inspiring and invigorating. I gave you an ultimate trial of determination, you see: if, after failing to solve the problem I'd set you – as, of course, I knew you would – you came back eager to learn more, to persist in your attempt for better or for worse, then I'd see you might have it in you to become a mathematician. But you… you weren't even curious enough to ask the solution! Indeed, you even gave me a signed declaration of your incompetence!'

The pent-up anger of many years now exploded. 'Do you know something, you old bastard? You may once have been a good mathematician, but as a human being you rate zero! Absolutely, totally zilch!'

To my surprise, this opinion was rewarded with a huge, hearty smile. 'On that, most favoured of nephews, I couldn't agree with you more!'

A month later I returned to the United States to prepare for my Senior year. I now had a new room-mate, unrelated to mathemarics. Sammy had meanwhile graduated and was at Princeton, already deeply involved in the problem that would in due course become his doctoral dissertation – with the exotic title: "The orders of the torsion subgroups of \Omega_{n} and the Adams spectral sequence'.

On my first free weekend I took the train and went to visit him. I found him quite changed, much more nervous and irritable than I had known him in the year of our association. He'd also acquired some kind of facial tic. Obviously, the torsion subgroups of \Omega_{n (whatever they were) had taken their toll on his nerves. We had dinner at a small pizza place across from the university and there I gave him a shortened version of Uncle Petros' story, as I'd heard it from him. He listened without once interrupting for question or comment.

After I was finished, he summed up his reaction in two words: 'Sour grapes.'

'What?'

'You should know – Aesop was a Greek.'

'What's Aesop got to do with it?'

'Everything. The fable of the fox who couldn't reach a yummy bunch of grapes and therefore decided they were unripe anyway. What a wonderful excuse your uncle found for his failure: he put the blame on Kurt Gödel! Wow!' Sammy burst out laughing. 'Audacious! Unheard of! But I have to grant it to him, it is original; in fact it's unique, it should go into some book of records! Never before has there been a mathematician seriously attributing his failure to find a proof to the Incompleteness Theorem!'

Although Sammy's words echoed my own first doubts, I lacked the mathematical knowledge to understand this immediate verdict.

'So, you think it's impossible that Goldbach's Conjecture is unprovable?'

'Man, what does "impossible" mean in this context?' Sammy sneered. 'As your uncle correctly told you, there is, thanks to Turing, no way of telling with certainty that a statement is a priori unprovable. But if mathematicians involved in advanced research started invoking Gödel, no one would ever go near the interesting problems – you see, in mathematics the interesting is always difficult. The Riemann Hypothesis has not yielded to proof after more than a Century? A case of application of the Incompleteness Theorem! The Four Colour Problem? Likewise! Fermat's Last Theorem still unproved? Blame it on evil Kurt Gödel! No one would ever have touched Hilbert's Twenty-three Problems; [14] indeed it's conceivable that all mathematical research, except the most trivial, would come to an end. Abandoning the study of a particular problem because it might be unprovable is like… like…' His face lit up when he found the appropriate analogy. 'Why, it's like not going out in the street for fear that a brick might fall on your head and kill you!'

'Let's face it,' he concluded, 'your Uncle Petros simply and plainly failed to prove Goldbach's Conjecture, like many greater men before him. But because, unlike them, he had spent his whole creative life on the problem, admitting his failure was unbearable. So, he concocted for himself this far-fetched, extravagant justification.'

Sammy raised his soda-glass in a mock toast. 'Here's to far-fetched excuses,' he said. Then he added in a more serious tone: 'Obviously, for Hardy and Littlewood to have accepted him as a collaborator, your uncle must have been a gifted mathematician. He could have made a great success of his life. Instead, he wilfully chose to throw it away by setting himself an unattainable goal and going after a notoriously difficult problem. His sin was Pride: he presumed that he would succeed where Euler and Gauss had failed.'

I was laughing now.

'What's so funny?' asked Sammy.

'After all these years of grappling with the mystery of Uncle Petros,' I said, ‘I’m back to square one. You just repeated my father's words, which I high-handedly rejected as philistine and coarse in my adolescence: "The secret of life, my son, is to set yourself attainable goals." It's exactly what you are saying now. That he didn't do so is, indeed, the essence of Petros' tragedy!'

Sammy nodded. 'Appearances are after all deceptive,’ he said with mock solemnity. 'It turns out the wise elder in the Papachristos family is not your Uncle Petros!'

I slept on the floor of Sammy's room that night, to the familiar sound of his pen scratching on paper accompanied by the occasional sigh or groan, as he struggled to untangle himself from the knots of a difficult topological problem. He left early in the morning to attend a seminar and in the afternoon we met at the Mathematics Library at Fine Hall, as arranged.

'We are going sightseeing,’ he said. 'I have a surprise for you.'

We walked a distance on a long suburban road lined with trees and strewn with yellow leaves.

'What courses are you taking this year?' Sammy asked as we walked towards our mysterious destination.

I started to list them: Introduction to Algebraic Geometry, Advanced Complex Analysis, Group Representation Theory…

'What about Number Theory?' he interrupted.

'No. Why do you ask?'

'Oh, I've been thinking about this business with your uncle. I wouldn't want you getting any crazy ideas into your head about following family tradition and tackling -'

I laughed.''Goldbach 's Conjecture? Not bloody likely!'

Sammy nodded. 'That's good. Because I have a suspicion that you Greeks are attracted to impossible problems.'

'Why? Do you know any others?'

'A famous topologist here, Professor Papakyriakopoulos. He's been struggling for years on end to prove the "Poincare Conjecture" – it's the most famous problem in low-dimensional topology, unproved for more than sixty years… ultra-hyper-difficult!'

I shook my head. 'I wouldn't touch anybody's famous unproved ultra-hyper-difficult problem with a ten-foot pole,’ I assured him.

‘I’m relieved to hear it,' he said.

We had reached a large nondescript building with extensive grounds. Once we had entered, Sammy lowered his voice.

'I got a special permit to come, in your honour,’ he said.

'What is this place?'

'You'll see.'

We walked down a corridor and entered a large, darkish room, with the atmosphere of a slightly shabby but genteel English gentlemen's club. About fifteen men, ranging from middle-aged to elderly, were seated in leather armchairs and couches, some by the windows, reading newspapers in the scanty daylight, others talking in little groups.

We settled ourselves at a little table in a corner.

'See that guy over there?' Sammy said in a low voice, pointing to an old Asian gentleman, quietly stirring his coffee.

'Yes?'

'He is a Nobel Prize in Physics. And that other one at the far end' – he indicated a plump, red-haired man gesturing heatedly as he spoke to his neighbour with a strong accent – 'is a Nobel Prize in Chemistry.' Then he directed my attention to two middle-aged men seated at a table near us. 'The one on the left is Andre Weil -'

'The Andre Weil?'

'Indeed, one of the greatest living mathematicians. And the other one with the pipe is Robert Oppenheimer – yes, the Robert Oppenheimer, the father of the atom bomb. He's the Director.'

'Director of what?'

'Of this place here. You are now in the Institute for Advanced Study, think-tank for the world's greatest scientific minds!'

I was about to ask more when Sammy cut me short. 'Shh! Look! Over there!'

A most odd-looking man had just come in through the door. He was about sixty, of average height and emaciated to an extreme degree, wearing a heavy overcoat and a knitted cap pulled down over his ears. He stood for a moment and peered at the room vaguely through extremely thick glasses. No one paid him any attention: he was obviously a regular. He made his way slowly to the tea and coffee table without greeting anybody, filled a cup with piain boiling water from the kettle and made his way to a seat by a window. He slowly removed his heavy overcoat. Underneath it he was wearing a thick jacket over at least four or five layers of sweaters, visible through his collar.

'Who is that man?' I whispered.

'Take a guess!'

'I haven't the slightest idea – he looks like a street person. Is he mad, or what?'

Sammy giggled. 'That, my friend, is your uncle's nemesis, the man who gave him the pretext for abandoning his mathematical career, none other than the father of the Incompleteness Theorem, the great Kurt Gödel!'

I gasped in amazement. 'My God! That's Kurt Gödel? But, why is he dressed like that?'

'Apparently he is convinced – despite his doctors' total disagreement – that he has a very bad heart and that unless he insulates it from the cold with all those clothes it will go into arrest.'

'But it's warm in here!'

'The modern high priest of Logic, the new Aristotle, disagrees with your conclusion. Which of the two should I believe, you or him?'

On our walk back to the university Sammy expounded his theory: ‘I think Gödel's insanity – for unquestionably he is in a certain sense insane – is the price he paid for coming too close to Truth in its absolute form. In some poem it says that "people cannot bear very much reality", or something like that. Think of the biblical Tree of Knowledge or the Prometheus of your mythology. People like him have surpassed the common measure; they've come to know more than is necessary to man, and for this hubris they have to pay.'

There was a wind blowing, lifting dead leaves in whirls around us. I sighed.


I’ll cut a long story (my own) short:

I never did become a mathematician, and this not because of any further scheming by Uncle Petros. Although his 'intuitive' depreciation of my abilities had definitely played a part in the decision by nurturing a constant, nagging sense of self-doubt, the true reason was fear.

The examples of the mathematical enfants terribles mentioned in my uncle's narrative – Srinivasa Ramanujan, Alan Turing, Kurt Gödel and, last but not least, himself – had made me think twice about whether I was indeed equipped for mathematical greatness. These were men who at twenty-five years of age, or even less, had tackled and solved problems of inconceivable difficulty and momentous importance. In this I'd definitely taken after my uncle: I didn't want to become a mediocrity and end up 'a walking tragedy', to use his own words. Mathematics, Petros had taught me, is a field that acknowledges only its greatest; this particular kind of natural selection offers failure as the only alternative to glory. Yet, hopeful as I still was in my ignorance about my abilities, it wasn't professional failure that I feared.

It all started with the sorry sight of the father of the Incompleteness Theorem padded with layers of warm clothing, of the great Kurt Gödel as a pathetic, deranged old man sipping his hot water in total isolation in the lounge of the Institute for Advanced Study.

When I returned to my university from the visit to Sammy, I looked up the biographies of the great mathematicians who had played a part in my uncle's story. Of the six mentioned in his narrative only two, a mere third, had lived a personal life that could be considered more or less happy and these two, significantly, were comparatively speaking the lesser men of the six, Caratheodory and Littlewood. Hardy and Ramanujan had attempted suicide (Hardy twice), and Turing had succeeded in taking his own life. Gödel's sorry state I've already mentioned. [15] Adding Uncle Petros to the list made the statistics even grimmer. Even if I still admired the romantic courage and persistence of his youth, I couldn't say the same of the way he'd decided to waste the second part of his life. For the first time I saw him for what he had clearly been all along, a sad recluse, with no social life, no friends, no aspirations, killing his time with chess problems. His was definitely not a prototype of the fulfilled life.

Sammy's theory of hubris had haunted me ever since I'd heard it, and after my brief review of mathematical history I embraced it wholeheartedly. His words about the dangers of coming too close to Truth in its absolute form kept echoing in my mind. The proverbial 'mad mathematician' was more fact than fancy. I came increasingly to view the great practitioners of the Queen of Sciences as moths drawn towards an inhuman kind of light, brilliant but scorching and harsh. Some couldn't stand it for long, like Pascal and Newton, who abandoned mathematics for theology.

Others had chosen haphazard, improvised ways out – Evariste Galois' mindless daring that led to his untimely death comes immediately to mind. Finally, some extraordinary minds had given way and broken down. Georg Cantor, the father of the Theory of Sets, led the latter part of his life in a lunatic asylum. Ramanujan, Hardy, Turing, Gödel and so many more were too enamoured of the brilliant light; they got too close, scorched their wings, fell and died.

In a short while I realized that even if I did have their gift (which, after listening to Uncle Petros' story, I began seriously to doubt) I definitely did not want to suffer their personal misery. Thus, with the Scylla of mediocrity on the one side and the Charybdis of insanity on the other, I decided to abandon ship. Although I did, come June, eventually get my BA in Mathematics, Ihad already applied for graduate studies in Business Economics, a field that does not traditionally provide material for tragedy.

Yet, I hasten to add, I've never regretted my years as a mathematical hopeful. Learning some real mathematics, even my tiny portion of it, has been for me the most invaluable lesson of life. Obviously, everyday problems can be handled perfectly well without knowledge of the Peano-Dedekind Axiomatic System, and mastery of the Classification of Finite Simple Groups is absolutely no guarantee of success in business. On the other hand, the non-mathematician cannot conceive of the joys that he's beert denied. The amalgam of Truth and Beauty revealed through the understanding of an important theorem cannot be attained through any other human activity, unless it be (I wouldn't know) that of mystical religion. Even if my education was meagre, even if it meant no more than getting my toes wet on the beach of the immense ocean of mathematics, it has marked my life for ever, giving me a small taste of a higher world. Yes, it has made the existence of the Ideal slightly more believable, even tangible.

For this experience I am forever in Uncle Petros' debt: it's impossible I would have made the choice without him as my dubious role model.

My decision to abandon plans of a mathematical career came as a joyful surprise to my father (the poor man had fallen into deep despair during my last undergraduate years), a surprise made even happier when he learned I would be going to business school. When, having completed my graduate studies and military service, I joined him in the family business, his happiness was at last complete.

Despite this volte-face (or maybe because of it?) my relationship with Uncle Petros blossomed anew after I returned to Athens, every vestige of bitterness on my part totally dissipated. As I gradually settled down to the routines of work and family life, visiting him became a frequent habit, if not a necessity. Our contact was an invigorating antidote to the increasing grind of the real world. Seeing him helped me keep alive that part of the self that most people lose, or forget about, with adulthood – call it the Dreamer or the Wonderer or simply the Child Within. On the other hand, I never understood what my friendship offered him, if we exclude the companionship he claimed not to need.

We wouldn't talk all that much on my visits to Ekali, as we'd found a means of communication better suited to two ex-mathematicians: chess. Uncle Petros was an excellent teacher and soon I came to share his passion (though unfortunately not his talent) for the game.

In chess, I also had the first direct experience of him as a thinker. As he analysed for my benefit the classic great games, or the more recent contests of the world's best players, I was filled with admiration for the workings of his brilliant mind, its immediate grasp of the most complex problems, its analytical power, the flashes of insight. When he confronted the board his features became fixed in utter concentration, his gaze became sharp and penetrating. Logic and intuition, the instruments with which he'd pursued for two decades the most ambitious intellectual dream, sparkled in his deep-set blue eyes.

Once, I asked him why he had never entered official competition.

He shook his head. 'Why should I strive to become a mediocre professional when I can bask in my status as an exceptional amateur?' he said. 'Besides, most favoured of nephews, every life should progress according to its basic axioms and chess wasn't among mine – only mathematics.'

The first time I ventured to ask him again about his research (after the extensive account of his life he had given me, we'd never again mentioned anything mathematicaL both of us apparently preferring to let our sleeping dogs lie) he immediately dismissed the matter.

'Let bygones be bygones and tell me what you see on the chessboard. It's a recent game between Petrosian and Spassky, a Sicilian Defence. White takes Knight to f4…'

More oblique attempts didn't work either. Uncle Petros would not be coaxed into another mathematical discussion – period. Whenever I attempted a direct mention it would always be: 'Let's stick to chess, shall we?'

His refusals, however, didn't make me give up.

My wish to draw him once again to the subject of his life's work was not fired by mere curiosity. Although it was a long time since I had any news of my old friend Sammy Epstein (last time I'd heard of him he was an assistant professor in California), I couldn't forget his explanation of Uncle Petros giving up his research. In fact, I'd come to invest it with great existential significance. The development of my own affair with mathematics had taught me an important lesson: one should be brutally honest with oneself about weaknesses, acknowledge them with courage and chart further course accordingly. For myself I had done this, but had Uncle Petros?

These were the facts: a) From an early age he had chosen to invest all his energy and time in an incredibly, but most probably not impossibly, difficult problem, a decision which I still continued to regard as basically noble; b) As might reasonably have been expected (by others, if not by himself) he had not achieved his goal; c) He had blamed his failure on the incompleteness of mathematics, deeming Goldbach's Conjecture unprovable.

Of this much I was now certain: the validity of his excuse had to be judged by the strict standards of the trade and, according to these, I accepted Sammy Epstein's opinion as final – a final verdict of unprovability a la Kurt Gödel is just not an acceptable conclusion of the attempt to prove a mathematical statement. My old friend's explanation was much closer to the point. It wasn't because of his 'bad luck' Uncle Petros hadn't managed to achieve his dream. The appeal to the Incompleteness Theorem was indeed a sophisticated form of 'sour grapes', meant only to shelter him from the truth.

With the passing of the years, I had learned to recognize the profound sadness that pervaded my uncle's life. His absorption in gardening, his kindly smiles or his brilliance as a chess player couldn't disguise the fact that he was a broken man. And the closer to him I got, the more I realized that the reason for his condition lay in his profound insincerity. Uncle Petros had lied to himself about the most crucial event in his life and this lie had become a cancerous growth that stifled his essence, eating away at the very roots of his psyche. His sin, indeed, had been Pride. And the pride was still there, nowhere more apparent than in his inability to come face to face with himself.

I've never been a religious man, yet I believe there is great underlying wisdom in the ritual of Absolution: Petros Papachristos, like every human being, deserved to end his life unburdened of unnecessary suffering. In his case, however, this had the necessary prerequisite of his admitting the mea culpa of his failure.

The context here not being religious, a priest could not do the job.

The only person fit to absolve Uncle Petros was I myself, for only I had understood the essence of his transgression. (The pride inherent in my own assumption I did not realize until it was too late.) But how could I absolve him if he did not first confess? And how could I lead him to confession unless we started once again to talk mathematics, a thing he persistently refused to do?

In 1971, I found unexpected assistance in my task.

The military dictatorship that then ruled the country, in a campaign to appear as a benevolent patron of culture and science, proposed to award a 'Gold Medal of Excellence' to a number of rather obscure Greek scholars who had distinguished themselves abroad. The list was short, since most of the prospective honourees, forewarned of the impending distinction, had hastened to exclude themselves; but topmost in it was 'the great mathematician of international fame, Professor Petros Papachristos'.

My father and Uncle Anargyros, in a totally uncharacteristic frenzy of democratic passion, strove to convince him to turn down this dubious honour. Talk of 'that old fool becoming the junta's lackey', 'giving the colonels an alibi', etc., filled our business offices and family homes. At moments of greater honesty the two younger brothers (both old men, by now) confessed to a less noble motive: the traditional reluctance of the businessman to be too closely identified with one political faction for fear of what will happen when another comes to power. Yet I, an experienced Papachristos family observer, could also discern a strong need for them to be proved right in their negative evaluation of his life, also tinged with an element of envy. Father's and Uncle Anargyros' world-view had always been founded on the simple premise that Uncle Petros was bad and they good, a black-and-white cosmology that distinguished between the grasshoppers and the ants, the dilettantes and 'responsible men'. It didn't sit at all well with them that the country's official government, Junta or no Junta, should honour 'one of life's failures', when the only rewards they ever got for their labours (labours, mind you, that also put food on his table) were financial.

I, however, took a different position. Beyond my belief that Uncle Petros deserved the honour (he did, after all, rate some recognition of his life's work, even if it came from the colonels) I had an ulterior motive. So I went to Ekali and, exercising to the full my influence as 'most favoured of nephews', convinced him to overcome his brothers' hypocritical appeals to democratic duty as well as his own misgivings and accept his Gold Medal of Excellence.

The award ceremony – that 'ultimate familial disgrace', according to Uncle Anargyros the late-blooming radical – was held in the main auditorium of the University of Athens. The Rector of the School of Physics and Mathematics, in his ceremonial robes, gave a short lecture on Uncle Petros' contribution to science. Predictably enough he referred almost exclusively to the Papachristos Method for the Solution of Differential Equations, which he lauded with elaborate rhetorical effusions. Still, I was agreeably surprised to hear him also make passing reference to Hardy and Littlewood and their 'appealing to our great fellow-countryman for assistance with their most difficult problems'. While all this was being propounded I stole side-glances at Uncle Petros and saw him blushing red with shame again and again, all the time retreating further into the throne-like, gilded armchair where they had him installed. The Prime Minister (the arch-dictator) then bestowed the Gold Medal of Excellence and afterwards there was a short reception, during which my poor uncle was required to pose for photographs with all the top brass of the Junta. (I have to confess that at this stage of the ceremony I feit a slight dose of guilt about the defining role I had played in his acceptance of the honour.)

When it was all over, he asked me to go back home for some chess, 'for purposes of recovery'. We started a game. I was a good enough player by that time to offer him decent resistance but not so good as to hold his interest after the ordeal he'd been through.

'What did you think of that circus?' he asked me, finally looking up from the board.

'The award ceremony? Oh, it was a bit boring, but I'm still glad you went through with it. Tomorrow it will be in all the newspapers.'

'Yes,’ he said, 'how the Papachristos Method for the Solution of Differential Equations is almost on a par with Einstein's Theory of Relativity and Heisenberg's Uncertainty Principle, one of the crowning achievements of twentieth-century science… How that fool of a Rector carried on! Did you notice, by the way,’ he added with a sour smile, 'the pregnant silence following the "ooohs" and "aaahs" and "ts-ts-ts's" of admiration at my extreme youth when I made the "great discovery"? You could almost hear everybody wondering: But how did the honouree spend the next fifty-five years of his life?'

Any sign of self-pity on his part bothered me inordinately.

'You know, Uncle,’ I provoked him, 'it's not anybody's fault but your own that people don't know of your work on Goldbach's Conjecture. How could they – you've never told! Had you ever written up a report of your research, things would be different.

The story of your quest itself would make a worth-while publication.'

'Yes,’ he sneered, 'a full footnote in Great Mathematical Failures of Our Century.'

'Well,' I mused, 'science advances by failures as well as successes. And anyway, it was a good thing your work in differential equations was acknowledged. I was proud to hear our family name associated with something other than money.'

Unexpectedly, a bright smile on his face, Uncle Petros asked me: 'Do you know it?'

'Do I know what?'

'The Papachristos Method for the Solution of Differential Equations?'

I'd been taken completely unawares and answered without thinking: 'No, I don't.'

His smile went away: 'Well, I expect they don't teach it anymore…'

I feit an upsurge of excitement – this was the chance I was waiting for. Although I had, in fact, ascertained while at university that the Papachristos Method was no longer taught (the advent of electronic calculation had rendered it obsolete), I lied to him, and with great vehemence: 'Of course they teach it, Uncle! It's just that I never took an elective in differential equations.'

'Get paper and pencil then, and I’ll tell you about it!'

I held back a triumphant cry. It was precisely what I'd hoped for when I had convinced him to accept the medal: that the honour might reawaken his mathematical vanity and rekindle his interest in his art, enough of it anyway to lure him into a discussion of Goldbach's Conjecture and beyond… to his real reason for abandoning it. Explaining to me the Papachristos Method was an excellent introduction.

I rushed to fetch paper and pencil before he changed his mind.

'You'll have to be a little patient,' he began. 'A lot of water has gone under the bridge since then. Let's see now,’ he murmured and began to scribble. 'Let us assume we have a partial differential equation in the Clairaut form… there! We now take…'

I followed his scribbles and explanations for almost an hour. Although I couldn't completely follow the argument, I showed exaggerated appreciation at every step.

'It's absolutely brilliant, Uncle!' I cried when he'd finished.

'Nonsense.' He brushed my praise aside, but I could see this modesty was not totally sincere. 'Sheer calculation of the grocery-bill variety, not real mathematics!'

The moment I was waiting for had arrived. 'Then talk to me about real mathematics, Uncle Petros. Talk to me about your work on Goldbach's Conjecture!'

He shot me a sideways glance, cunning, inquisitive and at the same time tentative. I held my breath.

'And what, if I may ask, is the purpose of your interest, Mr Almost-Mathematician?'

I had planned my answer to this beforehand, so as to put him in an emotional impasse.

'You owe it to me, Uncle! If for nothing eise, to compensate me for that summer of anguish in my sixteenth year, when I struggled for three months to prove it myself, floundering in my abysmal ignorance!'

He appeared to be considering this for a while, as if to make a point of not giving in all too easily. When he smiled I knew I had won.

'What exactly do you want to know about my work on Goldbach's Conjecture?'

I left Ekali after midnight with a copy of An Introduction to Number Theory by Hardy and Wright. (I had to prepare myself by learning 'some fundamentals’ he'd said.) I should point out to the non-specialist that mathematical books cannot normally be enjoyed like novels, in bed, in the bathtub, sprawled in an easy chair, or perched on the commode. To 'read' here means to understand, and for that you normally need a hard surface, paper, pencil and quality time. Since I had no intention of becoming a number theorist at the advanced age of thirty, I went through the Hardy-Wright book with only moderate attention ('moderate' in mathematics is 'considerable' by any other measure), without persisting on fully comprehending those details that resisted the initial assault. Even so, and taking into account that the study of the book was not my main occupation, it took me almost a month.

When I returned to Ekali, Uncle Petros, bless his soul, started to examine me as if I were a schoolchild.

'Have you read the whole book?'

'I have.'

'State Landau's Theorem.'

I did.

'Write out for me the proof of Euler 's Theorem of the \phi-function, the extension of Fermat's Little Theorem.'

I took paper and pencil and proceeded to do so, as best as I could.

'Now prove to me that the non-trivial zeros of the Riemann Zeta Function have real part equal to 1/2!'

I burst out laughing and he did too.

'Oh no, you don't!' I said. 'Not again, Uncle Petros! It's enough that you set me to prove Goldbach's Conjecture. Find somebody else to assign the Riemann Hypothesis!'

In the following two and a half months we had our ten 'Lessons on Goldbach's Conjecture', as he called them. What transpired in them is down on paper, with dates and times. Since I was now moving steadily towards the fulfilment of my main aim (his coming face to face with the reason for abandoning his research), I thought I'd also attain a secondary goal while at it: I kept meticulous notes so that, after his death, I could publish a short account of his Odyssey, perhaps an insignificant footnote to mathematical history, but still a worthy tribute to Uncle Petros – if not, alas, to his ultimate success, then certainly to his ingenuity and, more importantly, his dedication and single-minded persistence.

During the course of the lessons I witnessed an amazing metamorphosis. The mild, kindly, elderly gentleman I had known since my childhood, one easily mistaken for a retired civil servant, turned before my eyes into a man illuminated by a fierce intelligence and driven by an inner power of unfathomable depth. I'd caught small glimpses of this species of being before, during mathematical discussions with my old room-mate, Sammy Epstein, or even with Uncle Petros himself, when he sat before his chessboard. Listening to him unravel the mysteries of Number Theory, however, I experienced for the first and only time in my life the real thing. You didn't have to know mathematics to feel it. The sparkle in his eyes and an unspoken power emanating from his whole being were testimony enough. He was the absolute thorough-bred, pure unadulterated genius.

An unexpected fringe benefit was that the last remaining trace of ambivalence (apparently it had been there, dormant, all those years) regarding the wisdom of my decision to abandon mathematics was now dispelled. Watching my uncle do mathematics was enough to confirm it to the full. I was not made of the same mettle as he – this I realized now beyond the shadow of a doubt. Faced with the incamation of what I definitely was not, I accepted at last the truth of the dictum: Mathematicus nascitur non fit. The true mathematician is born, not made. I had not been born a mathematician and it was just as well that I had given up.

The exact content of the ten lessons is not within the scope of our story and I won't even attempt to refer to it. What matters here is that by the eighth we had covered the course of the initial period of Uncle Petros' research on Goldbach's Conjecture, culminating in his brilliant Partitions Theorem, now named after the Austrian who rediscovered it; also his other main result, attributed to Ramanujan, Hardy and Littlewood. In the ninth lesson he explained to me as much as I could understand of his rationale for changing the course of his attack from the analytic to the algebraic.

For the next he had asked me to bring along two kilos of lima beans. In fact, he had initially asked for navy beans, but then corrected himself, smiling sheepishly: 'Actually make it lima, so I can see them better. I'm not getting any younger, most favoured of nephews.'

As I drove to Ekali for the tenth (which, although I didn't know it yet, would be the last) lesson, I felt apprehensive: I knew from his narrative that he had given up precisely while working with the 'famous bean method'. Very soon, even in that imminent lesson, we would be reaching the cruciai point, his hearing of Gödel's Theorem and the end of his efforts to prove Goldbach's Conjecture. It would be then that I would have to launch my attack on his dearly held defences and expose his rationalization about unprovability for what it was: a mere excuse.

When I got to Ekaii he led me without a word to his socalled Irving room, which I found transformed. He'd pushed back what furniture there was against the walls, including even the armchair and the small table with the chessboard, and piled even higher piles of books along the perimeter, to create a wide, empty area in the centre. Without so much as a word he took the bag from my hands and started to arrange the beans on the floor, in a number of rectangles. I watched silently.

When he had finished he said: 'During our previous lessons we went over my early approach to the Conjecture. In this I had done good, perhaps even excellent, mathematics – but mathematics, nevertheless, of a rather traditional variety. The theorems I had proved were difficult and important, but they followed and extended lines of thought started by others, before me. Today, however, I will present to you my most important and original work, a ground-breaking advance. With the discovery of my geometric method I finally entered virgin, unexplored territory.'

'All the more pity that you abandoned it,’ I said, preparing the climate from the start for a confrontation.

He disregarded this and continued: 'The basic premise behind the geometric approach is that multiplication is an unnatural operation.'

'What on earth do you mean by unnatural?' I asked.

'Leopold Kronecker once said: "Our dear God made the integers, everything else is the work of man." Well, in the same way he made the integers, I think Kronecker forgot to add, the Almighty created addition and subtraction, or give and take.'

I laughed. 'I thought I came here for lessons in mathematics, not theology!'

Again he continued, ignoring the interruption. 'Multiplication is unnatural in the same sense as addition is natural. It is a contrived, second-order concept, no more really than a series of additions of equal elements. 3x5, for example, is nothing more than 5+5+5. To invent a name for this repetition and call it an 'operation' is the devil's work more likely…'

I didn't risk another facetious comment.

'If multiplication is unnatural,' he continued, 'more so is the concept of "prime number" that springs directly from it. The extreme difficulty of the basic problems related to the primes is in fact a direct outcome of this. The reason there is no visible pattern in their distribution is that the very notion of multiplication – and thus of primes – is unnecessarily complex. This is the basic premise. My geometric method is motivated simply by the desire to construct a natural way of viewing the primes.'

Uncle Petros then pointed at what he'd made while he was talking. 'What is that?' he asked me.

'A rectangle made of beans,' I replied. 'Of 7 rows and 5 columns, their product giving us 35, the total number of beans in the rectangle. All right?'

He proceeded to explain how he was struck by an observation which, although totally elementary, seemed to him to have great intuitive depth. Namely, that if you constructed, in theory, all possible rectangles of dots (or beans) this would give you all the integers – except the primes. (Since a prime is never a product, it cannot be represented as a rectangle but only as a single row.) He went on to describe a calculus for operations among the rectangles and gave me some examples. Then he stated and proved some elementary theorems.

After a while I began to notice a change in his style. In our previous lessons he'd been the perfect teacher, varying the tempo of his exposition in inverse proportion to its difficulty, always making sure I had grasped one point before proceeding to the next. As he advanced deeper into the geometric approach, however, his answers became hurried, fragmented and incomplete to the point of total obscurity. In fact, after a certain point my questions were ignored and what might have appeared at first as explanations I recognized now as overheard fragments of his ongoing infernal monologue.

At first, I thought this anomalous form of presentation was a result of his not remembering the details of the geometric approach as clearly as the more conventional mathematics of the analytic, and making desperate efforts to reconstruct it.

I sat back and watched him: he was walking about the living room, rearranging his rectangles, mumbling to himself, going to the mantelpiece where he'd left paper and pencil, scribbling, looking something up in a tattered notebook, mumbling some more, returning to his beans, looking here and there, pausing, thinking, doing some more rearranging, then scribbling some more… Increasingly, references to a 'promising line of thought', 'an extremely elegant lemma' or a 'deep little theorem' (all his own inventions, obviously) made his face light up with a self-satisfied smile and his eyes sparkle with boyish mischievousness. I suddenly realized that the apparent chaos was nothing eise than the outer form of inner, bustling mental activity. Not only did he remember the 'famous bean method' perfectly well – its memory made him positively gloat with pride!

A previously unthought-of possibility quickly entered my mind, only to become a near conviction moments later.

When first discussing Uncle Petros' abandoning Goldbach's Conjecture with Sammy, it had seemed obvious to both of us that the reason was a form of burnout, an extreme case of scientific battle fatigue after years and years of fruitless attacks. The poor man had striven and striven and striven and, after failing each time, was finally too exhausted and too disappointed to continue, Kurt Gödel providing him with a convenient if far-fetched excuse. But now, watching his obvious exhilaration as he played around with his beans, a new and much more exciting scenario presented itself: was it possible that, in direct contrast to what I'd thought until then, his surrender had come at the very peak of his achievement? In fact, precisely at the point when he felt he was ready to solve the problem?

In a flash of memory, the words he had used when describing the period just before Turing's visit came back – words whose real significance I had barely realized when I'd first heard them. Certainly he'd said that the despair and self-doubts he had felt in Cambridge, in that spring of 1933, had been stronger than ever. But had he not interpreted these as the 'inevitable anguish before the final triumph', even as the 'onset of the labour pains leading to the delivery of the great discovery'? And what about what he'd said a little earlier, just a little while ago, about this being his 'most important work', 'important and original work, a groundbreaking advance'? Oh my good God! Fatigue and disillusionment didn't have to be the causes: his surrender could have been the loss of nerve before the great leap into the unknown and his final triumph!

The excitement caused by this realization was such that I could no longer wait for the tactically correct moment. I launched my attack right away.

'I notice,’ I said, my tone accusing rather than observing, 'that you seem to think very highly of the "famous Papachristos bean method".'

I had interrupted his train of thought and it took a few moments for my comment to register.

' You have an amazing command of the obvious,’ he said rudely. 'Of course I think highly of it.'

'… in contrast to Hardy and Littlewood,’ I added, delivering my first seriousblow.

This brought the expected reaction – only to a much greater degree than I'd f oreseen.

'"Can't prove Goldbach with beans, old chap!"' he said in a gruff, boorish tone, obviously parodying Littlewood. Then, he took on the other member of the immortal mathematical pair in a cruel mimicry of effeminacy. "Too elementary for your own good, my dear fellow, infantile even!'"

He banged his fist on the mantelpiece, furious. "That ass Hardy,’ he shouted, 'calling my geometric method "infantile" – as if he understood the first thing about it!'

'Now, now, Uncle,’ I said scoldingly, 'you can't go calling G. H. Hardy an ass!'

He banged his fist again, with greater force.

'An ass he was, and a sodomite too! The "great G. H. Hardy" – the Queen of Number Theory!'

This was so untypical of him I gasped. 'My, my, we are getting nasty, Uncle Petros!'

'Not at all! I'll call a spade a spade and a bugger a bugger!'

If I was startled I was also exhilarated: a totally new man had magically appeared before my eyes. Could it be that, together with the 'famous bean method', his old (I mean his young) seif had at last resurfaced? Could I now be hearing, for the first time, Petros Papachristos' real voice? Eccentricity – even Obsession – was certainly more characteristic of the single-minded, overambitious, brilliant mathematician of his youth than the gentle, civilized manners I'd come to associate with my elderly Uncle Petros. Conceit and malice towards his peers could well be the necessary other side of his genius. After all, both were perfectly suited to his capital sin, as diagnosed by Sammy: Pride.

To push it to its limit I used a casual tone: 'G. H. Hardy's sexual inclinations do not concern me,' I said. 'All that is relevant, vis-ä-vis his opinion of your "bean method", is that he was a great mathematician!'

Uncle Petros' face went crimson. 'Bollocks,' he growled. 'Prove it!'

'I don't have to,' I said dismissively. 'His theorems speak for themselves.'

'Oh?Which one?'

I stated two or three of the results I remembered from his textbook.

'Ha!' Uncle Petros snarled. 'Mere calculations of the grocery-bill variety! But show me one great idea, one inspired insight… You can't? That's because there isn't one!' He was fuming now. 'Oh, and while you're at it, tell me of a theorem the old pansy proved on his own, without good old Littlewood or poor dear Ramanujan holding his hand – or whatever other part of his anatomy it was they were holding!'

The mounting nastiness signalled that we were approaching a breakthrough. A tiny extra bit of annoyance was probably all that was necessary to bring it about.

'Really, Uncle,’ I said, trying to sound as haughty as possible. This is beneath you. After all, whatever theorems Hardy proved, they were certainly more important than yours!'

'Oh yes?' he snapped back. 'More important than Goldbach's Conjecture?'

I burst into incredulous laughter, despite myself. 'But you didn't prove Goldbach's Conjecture, Uncle Petros!'

'I didn't prove it, but -'

He broke off in mid-sentence. His expression betrayed he'd said more than he wanted to.

'You didn't prove it but what?’ I pressed him. 'Come on, Uncle, complete what you were going to say! You didn't prove it but were very dose to it? I'm right – am I not?'

Suddenly, he stared at me as if he were Hamlet and I his father's ghost. It was now or never. I leapt up from my seat.

'Oh, for God's sake, Uncle,' I cried. ‘I’m not my father or Uncle Anargyros or grandfather Papachristos! I know some mathematics, remember? Don't give me that crap about Gödel and the Incompleteness Theorem! Do you think I swallowed for a single moment that fairy tale of your "intuition telling you the Conjecture was unprovable"! No – I knew it from the very start for what it was, a pathetic excuse for your failure. Sourgrapes!’

His mouth opened in wonder – from ghost I must have been transformed into a celestial vision.

'I know the whole truth, Uncle Petros,’ I continued fervently. 'You got to within a hair's breadth of the proof! You were almost there… Almost… All but the final step…' – my voice was coming out in a humming, deep chant -'… and then, you lost your nerve! You chickened out, Uncle dearest, didn't you? What happened! Did you run out of willpower or were you just too scared to follow the path to its ultimate conclusion? Whatever the case, you'd always known it deep inside: the fault is not with the Incompleteness of Mathematics!'

My last words had made him recoil and I thought I might as well play the part to the hilt: I grabbed him by the shoulders and shouted straight into his face.

'Face it, Uncle! You owe it to yourself, can't you see that? To your courage, to your brilliance, to all those long, fruitless, lonely years! The blame for not proving Goldbach's Conjecture is all your own – just as the triumph would have been totally yours if you'd succeeded! But you didn't succeed! Goldbach's Conjecture is provable and you knew that all along! It's just that you didn't manage to prove it! You failed -you failed, God damn it, and you've got to admit it, at last!'

I had run out of breath.

As for Uncle Petros, for a slight moment his eyes closed and he wavered. I thought that he was going to pass out, but no – he instantly came to, his inner turmoil now unexpectedly melting into a soft, mellow smile.

I smiled too: naively, I thought that my wild ranting had miraculously achieved its purpose. In fact, at that moment I would have made a bet that his next words would be something like: 'You are absolutely right. I failed. I admit it. Thank you for helping me do it, most favoured of nephews. Now, I can die happy'

Alas, what he actually said was: 'Will you be a good boy and go get me five more kilos of beans?'

I was stunned – all of a sudden he was the ghost and I Hamlet.

'We – we must finish our discussion first,' I faltered, too shocked for anything stronger.

But then he started pleading: 'Please! Please, please, please get me some more beans!'

His tone was so intolerably pathetic that my defences crumbled to dust. For better or for worse, I knew that my experiment in enforced self-confrontation had ended.

Buying uncooked beans in a country where people don't do their grocery shopping in the middle of the night was a worthy challenge to my developing entrepreneurial skills. I drove from taverna to taverna, beguiling the cooks into selling me from their pantry stock a kilo here, half a kilo there, until I accumulated the required quantity. (It was probably the most expensive five kilos of beans ever.)

When I got back to Ekali, it was past midnight. I found Uncle Petros waiting for me at the garden gate.

'You are late!' was his only greeting.

I could see that he was in a state of tremendous agitation.

'Everything all right, Uncle?'

'Are these the beans?'

'They are, but what's the matter? What are you so worked up about?'

Without answering he grabbed the bag. 'Thank you,' he said and began to close the gate.

'Shan't I come in?' I asked, surprised.

'It's too late,’ he said.

I was reluctant to leave him until I found out what was going on.

'We don't have to talk mathematics,’ I said. 'We can have a little game of chess or, even better, drink some herbal tea and gossip about the family.'

'No,’ he said with finality. 'Goodnight.' He walked fast towards his small house.

'When is the next lesson?' I shouted af ter him.

‘I’ll call you,’ he said, went in and banged the door behind him.

I remained standing on the pavement for a while, wondering what to do, whether to attempt once again to enter the house, to talk to him, to see if he was all right. But I knew he could be stubborn as a mule. Anyway, our lesson and my noctumal search for beans had drained me of all energy.

Driving back to Athens I was pestered by my conscience. For the first time, I questioned my course of action. Could my high-handed stance, supposedly intended to lead Uncle Petros into a therapeutic showdown, have been nothing more than my own need to get even, an attempt to avenge the trauma he'd inflicted on my teenage seif? And, even if that weren't so, what right did I have to make the poor old man face

the phantoms of his past, despite himself? Had I seriously considered the consequences of my inexcusable immaturity? The unanswered questions abounded, but still, by the time I got home I had rationalized myself out of the moral tight spot: the distress I'd obviously caused Uncle Petros had most probably been the necessary – the obligatory – step in the process of his redemption. What I'd told him was, after all, too much to digest at one go. Obviously the poor man only needed a chance to think things over in peace. He had to admit his failure to himself, before he could do so to me…

But if that was the case, why the extra five kilos of beans?

A hypothesis had begun to form in my mind, but it was too outrageous to be given serious consideration – until morning anyway.

Nothing in this world is truly new – certainly not the high dramas of the human spirit. Even when one such appears to be an original, on closer examination you realize it's been enacted before, with different protagonists, of course, and quite possibly with many variations in its development. But the main argument, the basic premise, repeats the same old story.

The drama played out during Petros Papachristos' final days is the last in a triad of episodes from the history of mathematics, unified by a single theme: the Mystery-solution to a Famous Problem by an Important Mathematician. [16]

By majority consent, the three most famous unsolved mathematical problems are: (a) Fermat's Last Theorem, (b) the Riemann Hypothesis and (c) Goldbach's Conjecture.

In the case of Fermat's Last Theorem, the mystery-solution existed from its first statement: in 1637, while he was studying Diophantus' Arithmetica, Pierre de Fermat made a note in the margin of his personal copy, right next to proposition II.8 referring to the Pythagorean theorem, in the form x^2 + y^2 = z^2. He wrote: ‘It is impossible to separate a cube into two cubes, or a biquadrate (fourth power) into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which, however, this margin is not large enough to contain.'

After the death of Fermat his son collected and published his notes. A thorough search of his papers, however, failed to reveal the demonstratio mirabilis, the 'marvellous proof’ that his father claimed to have found. Equally in vain have mathematicians ever since sought to rediscover it. [17] As for the verdict of history on the existence of the mystery-solution: it's ambiguous. Most mathematidans today doubt that Fermat indeed had a proof. The worst-case theory has it that he was consciously lying, that he had not verified his guess and his margin-note was mere bragging. What's likelier, however, is that he was mistaken, the demonstratio mirabilis crippled by an undetected fault.

In the case of the Riemann Hypothesis, the mystery-solution was in fact a metaphysical practical joke, with G. H. Hardy as its perpetrator. This is how it happened:

Preparing to board a cross-Channel ferry during a bad storm, the confirmed atheist Hardy sent off to a colleague a postcard with the message: 'I have the proof to the Riemann Hypothesis.' His reasoning was that the Almighty, whose sworn enemy he was, would not permit him to reap such an exalted undeserved reward and would therefore see to his safe arrival, in order to have the falsity of his claim exposed.

The mystery-solution of Goldbach's Conjecture completes the triad.

On the morning after our last lesson, I telephoned Uncle Petros. At my insistence, he had recently agreed to have a line installed, on the condition that only I, and no one eise, would know the number.

He answered sounding tense and distant. 'What do you want?'

'Oh, I just called to say hello,' I said. 'Also to apologize. I think I was unnecessarily rude last night.'

There was a pause.

'Well,’ he said, 'actually I'm busy at the moment. Why don't we talk again… shall we say next week?'

I wanted to assume that his coldness was due to the fact that he was upset with me (as he had every reason to be, after all) and merely expressing his resentment. Still, I feit a nagging unease.

'Busy with what, Uncle?' I persisted.

Another pause.

'I-I'll tell you about it some other time.'

He was obviously eager to hang up so, before he could cut me off, I impulsively blurted out the suspicion that had taken shape during the night.

'You wouldn't by any chance have resumed your researches, would you, Uncle Petros?'

I heard a sharp intake of breath. 'Who – who told you that?' he said hoarsely.

I tried to sound casual. 'Oh, come on, give me some credit for having come to know you. As if it needed telling!'

I heard the click of his hanging up. My God – I was right! The crazy old fool had gone off his rocker. He was trying to prove Goldbach's Conjecture!

My guilty conscience stung me. What had I done? Humankind indeed cannot stand very much reality – Sammy's theory of Kurt Gödel's insanity also applied, in a different way, to Uncle Petras. I had obviously pushed the poor old man to his uttermost limit and then beyond it. I'd aimed straight at his Achilles heel and hit it. My ridiculous simple-minded scheme to force him into self-confrontation had destroyed his fragile defences. Heedlessly, irresponsibly, I had robbed him of the carefully nurtured justification of his failure: the Incompleteness Theorem. But I had put nothing in its place to sustain his shattered self-image. As his extreme reaction now showed, the exposure of his failure (to himself, more than to me) had been more than he could bear. Stripped of his cherished excuse he had taken, of necessity, the only way left for him to go: madness. For what else was the endeavour to search, in his late seventies, for the proof that he had failed to find when he was at the peak of his powers? If that wasn't total irrationality, what was?

I walked into my father's office filled with apprehension. Much as I hated to allow him into the charmed circle of my bond with Uncle Petros, I feit obliged to let him know what had happened. He was after all his brother, and any suspicion of serious illness was certainly a family matter. My father dismissed my self-recriminations about causing the crisis as so much poppycock. According to the official Papachristos world-view, a man had only himself to blame for his psychological condition, the only acceptable external reason for emotional discomfort being a serious drop in the price of stocks. As far as he was concerned, his older brother's behaviour had always been bizarre, and one more instance of eccentricity was definitely not to be taken seriously.

'In fact,’ he said, 'the condition you describe – absent-mindedness, self-absorption, abrupt changes of mood, irrational demands for beans in the middle of the night, nervous tics, etc. – reminds me of how he was carrying on when we visited him in Munich, back in the late twenties. Then, too, he was behaving like a madman. We'd be at a nice restaurant enjoying our wurst and he'd be squirming around as if there were nails in his chair, his face twitching like mad.'

'Quod erat demonstrandum,' I said. "That's exactly it. He's back doing mathematics. In fact, he's back working on Goldbach's Conjecture – ridiculous as that may sound at his age.'

My father shrugged. 'It's ridiculous at any age,’ he said. 'But why worry? Goldbach's Conjecture has already done him all the harm possible. Nothing worse can come of it.'

But I wasn't so sure about that. In fact, I was quite certain that a lot worse things could be in store for us. Goldbach's resurrection was bound to stir up unfulfilled passions, to aggravate deep-buried, terrible, unhealed wounds. His absurd new application to the old problem boded no good.

After work that evening, I drove to Ekali. The ancient VW beetle was parked outside the house. I crossed the front yard and rang the bell. There was no response, so I shouted: 'Open up, Uncle Petros; it's me!'

For a few moments I feared the worst, but then he appeared at a window and stared vaguely in my direction. There was no sign of his usual pleasure at seeing me, no surprise, no greeting – he just stared.

'Good afternoon,’ I said. 'I just came by to say hello.'

His normally serene face, the face of a stranger to life's usual worries, was now marked by extreme tension, his skin pale, his eyes red with sleeplessness, his brow furrowed with concern. He was also unshaven, the first time I'd seen him so. His stare continued absent, unfocused. I wasn't even sure he knew who I was.

'Come on, Uncle dear, please open up for the most favoured,’ I said with a fatuous smile.

He disappeared and after a while the door creaked open. He stood there, blocking my entry, wearing his pyjama bottoms and a wrinkled vest. It was evident he didn't want me to enter.

'What's wrong, Uncle?' I asked. ‘I’m worried about you.'

'Why should you be worried?' he said, now forcing himself to sound normal. 'Everything's fine.'

'Are you sure?'

'Of course I'm sure.'

Then, with a snappy gesture, he beckoned me closer. After quickly, anxiously glancing around, he leaned towards me, his lips almost touching my ear, and whispered: 'I saw them again.'

I didn't understand. 'Who did you see?'

'The girls! The twins, the number 2^100!'

I remembered the strange apparitions of his dreams.

'Well,' I said, trying to sound as casual as possible. ‘If you are once again involved with mathematical research, you are once again having mathematical dreams. Nothing strange about that…'

I wanted to keep him talking so as to (figuratively, but if need be also literally) put a foot in the door. I had to get some sense of how bad his condition was.

'So what happened, Uncle,' I asked, feigning great interest in the matter. 'Did the girls speak to you?'

'Yes,’ he said, 'they gave me a…' His voice quickly trailed off, as if he was afraid he'd said too much.

'A what?' I asked. 'A clue?'

He became suspicious again. 'You mustn't tell,’ he said sternly.

'Mum's the word,' I said.

He had started to close the door. Convinced now that his situation was extremely serious and that the time had come for emergency action, I grasped the knob and started to push. As he felt my force, he tensed up, gritted his teeth and struggled to prevent me from entering, his face contorted to a grimace of desperation. Fearing the effort might be too much for him (he was nearing eighty, after all) I reduced the pressure a bit for a final attempt at reason.

Of all the possible stupid things I could have said to him, I chose this: 'Remember Kurt Gödel, Uncle Petros! Remember the Incompleteness Theorem – Goldbach's Conjecture is unprovable!'

Instantly, his expression changed from despair to wrath. 'Fuck Kurt Gödel,' he barked, 'and fuck his Incompleteness Theorem!' With an unexpected upsurge of strength, he overcame my resistance and slammed the door shut in my face.

I rang the bell again and again, banged the door with my fist and shouted. I tried threats, reasoning and pleading, but nothing worked. When a torrential October rain began to fall I hoped that, mad or not, Uncle Petros might be moved by mercy and let me in. But he wasn't. I left, soaking wet and very worried.

From Ekali I drove straight to our family doctor and explained the Situation. Without altogether ruling out serious mental disturbance (possibly triggered by my unwarranted interference in his defence mechanisms) he suggested two or three organic problems as likelier causes of my uncle's transformation. We decided to go to his house first thing the next morning, force our way in if necessary, and submit him to a thorough medical examination.

That night I couldn't sleep. The rain was getting stronger, it was past two o'clock and I was sitting at home hunched in front of the chessboard, just as Uncle Petros must have been on innumerable sleepless nights, studying a game from the recent world championship. Yet my concern kept interfering and I couldn't concentrate.

When I heard the ringing I knew it was he, even though he'd never yet initiated a call on his newly installed telephone.

I jumped up and answered.

'Is that you, Nephew?' He was obviously all worked up about something.

'Of course it's me, Uncle. What's wrong?'

'You must send me someone. Now!'

I was alarmed. '"Someone"? A doctor you mean?'

'What use would a doctor be? A mathematician, of course!'

I humoured him: 'I'm a mathematician, Uncle; I’ll come right away! Just promise to open the door, so I won't catch pneumonia and -'

He obviously didn't have time for irrelevancies. 'Oh hell!' he grunted and then: 'All right, all right, you come, but bring another one as well!'

'Another mathematician?'

'Yes! I must have two witnesses! Hurry!'

'But why do the witnesses have to be mathematicians?'

Naively, I had thought at first he wanted to write his will.

'To understand my proof!'

‘Proof of what?’

'Goldbach's Conjecture, you idiot – what else!'

I chose my next words very carefully. 'Look, Uncle Petros,' I said, 'I promise to be with you as soon as my car will get me there. Let's be reasonable, mathematicians aren't kept on call – how on earth can I get one at two o'clock in the morning? You'll tell me all about your proof tonight and tomorrow we will go together -'

But he cut me off, screaming. 'No, no, no! There's no time for any of that! I need my two witnesses and I need them now!’ Then he broke down and started sobbing. 'O nephew, it's so… it's so…'

'So what, Uncle? Tell me!'

'Oh, it's so simple, so simple, my dearest boy! How is it possible that all those years, those endless years, I hadn't realized how blessedly simple it was!'

I cut him off. ‘I’ll be there as soon as I can.'

'Wait! Wait! Waaaaa-it!!!' He was now in panic. 'Swear you won't come alone! Get the other witness! Hurry… Hurry up, I implore you! Get the witness! There's no time!'

I tried to appease him: 'Oh, come on, Uncle; there can't be such a rush. The proof won't go away, you know!'

These were his last words: 'You don't understand, dear boy – there's no time left!' His voice then dropped to a low, conspiratorial whisper, as if he didn't want to be overheard by someone close by: 'You see, the girls are here. They are waiting to take me.'

By the time I arrived in Ekali, breaking all speed records, it was too late. Our family doctor (I had picked him up on the way) and I found Uncle Petros' lifeless body slumped on the paving of his little terrace. The torso was leaning against the wall, the legs spread open, the head turned towards us as if in welcome. A flash of distant lightning revealed his features fixed in a wonderful smile of deep, absolute contentment – I imagine it was that which guided the doctor in his instant diagnosis of a stroke. All around him were hundreds of lima beans. The rain had destroyed their neat parallelograms and now they were scattered all over the wet terrace, sparkling like precious jewels.

The rain had just stopped and the air was infused with the invigorating smell of wet earth and pine trees.

Our last exchange over the telephone is the only evidence of Petros Papachristos' mystery-solution to Goldbach's Conjecture.

Unlike Pierre de Fermat's illustrious marginal note, however, it is extremely unlikely that my uncle's demonstratio mirabilis of his famous problem will tempt a host of mathematical hopefuls to attempt to reproduce it. (No rise in the price of beans is expected.) This is as it should be. Fermat's sanity was never in question; no one ever had reason to believe he was in anything less than total possession of his senses when he stated his Last Theorem. Unfortunately, the same cannot be said of my Uncle Petros. When he announced his triumph to me he was probably as mad as a harter. His last words were uttered in a state of terminal confusion, the total relinquishment of logic, the Night of Reason that dimmed the light of his final moments. It would thus be extremely unfair to have him posthumously declared a charlatan by attributing a serious intention to a declaration obviously made in a half-delirious state, his brain most probably already ravaged by the stroke that, a short while later, killed him.

So: did Petros Papachristos prove Goldbach's Conjecture in his final moments? The wish to protect his memory from any chance of ridicule obliges me to state it as unequivocally as possible: the official answer must be 'No'. (My own opinion need not concern mathematical history – I will therefore keep it to myself.)

The funeral was strictly family, with only a wreath and a single representative from the Hellenic Mathematical Society.

The epitaph later carved on Petros Papachristos' tomb, below the dates marking the limits of his earthly existence, was chosen by me, after I had overcome the initial objections of the family elders. They form one further addition to the collection of posthumous messages that make the First Cemetery of Athens one of the world's most poetic:


EVERY EVEN NUMBER GREATER THAN 2

IS THE SUM OF TWO PRIMES

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